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Students will learn to identify, write, and analyze the truth value of conditional statements. They will practice writing the inverse, converse, and contrapositive of a conditional statement. Vocabulary includes hypothesis, conclusion, negation, logically equivalent statements.
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Objectives Students will… Identify, write, and analyze the truth value of conditional statements. Write the inverse, converse, and contrapositive of a conditional statement.
Warm Up Determine if each statement is true or false. 1.The measure of an obtuse angle is less than 90°. 2. All perfect-square numbers are positive. 3. Every prime number is odd. 4. Any three points are coplanar. F T F T
Vocabulary conditional statement hypothesis conclusion truth value negation converse inverse contrapostive logically equivalent statements
By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion.
Example 1: Identifying the Parts of a Conditional Statement Identify the hypothesis and conclusion of each conditional. A. If today is Thanksgiving Day, then today is Thursday. Hypothesis: Today is Thanksgiving Day. Conclusion: Today is Thursday. B. A number is a rational number if it is an integer. Hypothesis: A number is an integer. Conclusion: The number is a rational number.
Writing Math “If p, then q” can also be written as “if p, q,” “q, if p,” “p implies q,” “q when p,” “q because p,” and “p only if q.” C. I sneeze, when I have a cold. Hypothesis: I have a cold. Conclusion: I sneeze. D. It snows because it is cold. Hypothesis: It is cold. Conclusion: It snows.
*Sentences without "if" and "then" can be written as conditionals. *Identify the hypothesis and conclusion by figuring out which part of the statement depends on the other.
Example 2A: Writing a Conditional Statement Write a conditional statement from the following. An obtuse triangle has exactly one obtuse angle. An obtuse triangle has exactly one obtuse angle. Identify the hypothesis and the conclusion. If a triangle is obtuse, then it has exactly one obtuse angle.
Example 2B: Writing a Conditional Statement Write a conditional statement from the following. If an animal is a blue jay, then it is a bird. The inner oval represents the hypothesis, and the outer oval represents the conclusion.
A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis is true and the conclusion is false. To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false.
Example 3A Determine if the conditional “If a number is odd, then it is divisible by 3” is true. If false, give a counterexample. An example of an odd number is 7. It is not divisible by 3. In this case, the hypothesis is true, but the conclusion is false. Since you can find a counterexample, the conditional is false.
Example 3B: Analyzing the Truth Value of a Conditional Statement Determine if the conditional is true. If false, give a counterexample. If two angles are acute, then they are congruent. You can have acute angles with measures of 80° and 30°. In this case, the hypothesis is true, but the conclusion is false. Since you can find a counterexample, the conditional is false.
Example 3C: Analyzing the Truth Value of a Conditional Statement Determine if the conditional is true. If false, give a counterexample. If an even number greater than 2 is prime, then 5 + 4 = 8. An even number greater than 2 will never be prime, so the hypothesis is false. 5 + 4 is not equal to 8, so the conclusion is false. However, the conditional is true because the hypothesis is false.
*The negation of statement p is “not p,” written as ~p. *The negation of a true statement is false, and the negation of a false statement is true.
Check It Out! Example 4 Write the converse, inverse, and contrapostive of the conditional statement “If an animal is a cat, then it has four paws.” Find the truth value of each. If an animal is a cat, then it has four paws.
Check It Out! Example 4 If an animal is a cat, then it has four paws. Converse:If an animal has 4 paws,then it is a cat. There are other animals that have 4 paws that are not cats, so the converse is false. Inverse: If an animal is not a cat, thenit does not have 4 paws. There are animals that are not cats that have 4 paws, so the inverse is false. Contrapositive: If an animal does not have 4 paws, thenit is not a cat; True. Cats have 4 paws, so the contrapositive is true.
*Related conditional statements that have the same truth value are called logically equivalent statements. A conditional and its contrapositive are logically equivalent, and so are the converse and inverse.
Helpful Hint The logical equivalence of a conditional and its contrapositive is known as the Law of Contrapositive.
Lesson Quiz: Part I Identify the hypothesis and conclusion of each conditional. 1. A triangle with one right angle is a right triangle. 2. All even numbers are divisible by 2. 3. Determine if the statement “If n2 = 144, then n = 12” is true. If false, give a counterexample. H: A triangle has one right angle. C: The triangle is a right triangle. H: A number is even. C: The number is divisible by 2. False; n = –12.
Lesson Quiz: Part II Identify the hypothesis and conclusion of each conditional. 4. Write the converse, inverse, and contrapositive of the conditional statement “If Maria’s birthday is February 29, then she was born in a leap year.” Find the truth value of each. Converse: If Maria was born in a leap year, then her birthday is February 29; False. Inverse: If Maria’s birthday is not February 29, then she was not born in a leap year; False. Contrapositive: If Maria was not born in a leap year, then her birthday is not February 29; True.
Check It Out! Example 2 Write a conditional statement from the sentence “Two angles that are complementary are acute.” Identify the hypothesis and the conclusion. Two angles that are complementary are acute. If two angles are complementary, then they are acute.
Check It Out! Example 1 Identify the hypothesis and conclusion of the statement. "A number is divisible by 3 if it is divisible by 6." Hypothesis: A number is divisible by 6. Conclusion: A number is divisible by 3.
Example 3A: Analyzing the Truth Value of a Conditional Statement Determine if the conditional is true. If false, give a counterexample. If this month is August, then next month is September. When the hypothesis is true, the conclusion is also true because September follows August. So the conditional is true.
Example 4: Biology Application Write the converse, inverse, and contrapositive of the conditional statement. Use the Science Fact to find the truth value of each. If an animal is an adult insect, then it has six legs.
Example 4: Biology Application If an animal is an adult insect, then it has six legs. Converse: If an animal has six legs, then it is an adult insect. No other animals have six legs so the converse is true. Inverse:If an animal is not an adult insect,then it does not have six legs. No other animals have six legs so the converse is true. Contrapositive:If an animal does not have six legs,thenit is not an adult insect. Adult insects must have six legs. So the contrapositive is true.
